IOAA 2018, Теория — Задача 10. Vega and Altair

• (a) 34.19582 deg

• (b) 4.4855 pc

• (c) Vega: θ1 = 35.0697°; Altair: θ2 = 54.302°

• (d) 2

• (e) α3 = 240.087° ≈ 16h0m21s, δ3 = −27.945°

• (f) Vega: 781200 BCE (about 783200 years ago); Altair: printed as 3617 BCE (about 363700 years ago)

  The official solution prints “number of years =363700 years” and then “It will happen in the year 3617 BCE”; the two printed figures are inconsistent (363700 years before 2018 is about 361700 BCE).

• (g) 40.75 deg

• (h) No such point exists (the determinant is non-zero, −72.47).

(a) Assuming the positions of the two starts are: Vega: (α$_1$,δ$_1$); Altair : (α$_2$,δ$_2$)

$(\alpha_1,\delta_1) = (279.23538^{o}, 38.78547^{o})$ and $(\alpha_2,\delta_2) = (297.69875^{o}, 8.87036^{o})$ (2 points)

Cosine formula of the spherical geometry is given by:

$cos[a] = cos[b]cos[c] + sin[b]sin[c]cos[A]$ (0.5 point)

In our triangle, b = (90 – δ$_1$), c = (90 – δ$_2$) and A = (α$_2$ - α$_1$). Then the angular distance rd between Vega and Altair is:

$cos[\beta] = cos[90^{o} - \delta_1]cos[90^{o} - \delta_2] + sin[90^{o} - \delta_1]sin[90^{o} - \delta_2]cos[\alpha_2 - \alpha_1];$ (3 points)

$cos[\beta] = cos[51.21453^{o}]cos[81.12964^{o}] + sin[51.21453^{o}]sin[81.12964^{o}]cos[18.46337^{o}];$ (1.5 points)

$\beta = 34.19582^{o}$ (2 points)

(b) The distance of Vega is: $r_1 = 1/(130.23 \times 10^{-3}) = 7.6787pc$

The distance of Altair is: $r_2 = 1/(194.95 \times 10^{-3}) = 5.1295pc$ (2 points)

Using the Law of Cosines： $d^2 = {r_1}^2 + {r_2}^2 - 2r_1r_2cos[\beta]$ (2 point)

we can obtain the distance: $d = 4.4855pc$ (2 points)

(c) Given the proper motions and radial velocities of the two stars, the directions of their movements on the celestial sphere can be estimated.

For Vega:

(2 points)

For Altair:

(1 points)

(d) As the position angles of the proper motion of two stars is different, the stars’ paths will intersect. As the paths are circular on celestial sphere, they will intersect in exactly two points.

(2 points)

(e) Let the closer point of intersection be I with the coordinates (α$_3$,δ$_3$).

In triangle PVA, $PAV = arcsin\left[\frac{sinVPAsinPV}{sinVA}\right] = arcsin\left[\frac{sin(\alpha_1-\alpha_2)sin(90^{o}-\delta_1)}{sin\beta}\right] = 26.055^{o}$

$PVA = arcsin\left[\frac{sinVPAsinPA}{sinVA}\right] = arcsin\left[\frac{sin(\alpha_1-\alpha_2)sin(90^{o}-\delta_2)}{sin\beta}\right] = 146.196^{o}$ (5 points)

Note that angle PVA is more than 90º, which will be evident from the diagram. (1 point)

The motion of the stars on the celestial sphere will always be along some great circle. So VAI is a spherical triangle. Using four parts formula for VAI

(5 points)

by sine rule, $AI = arcsin\left[\frac{sinVIsinIVA}{sinVAI}\right] = arcsin\left[\frac{sin76.085^{o}sin68.900^{o}}{sin99.643^{o}}\right] = 66.715^{o}$

(2 points counted in next part)

Now we use triangle PVI

by cosine rule, $cos[PI] = cos[PV]cos[VI] + sin[PV]sin[VI]cos[PVI];$

$cos[90^{o} - \delta_3] = cos[90^{o} - \delta_1]cos[VI] + sin[90^{o} - \delta_1]sin[VI]cos[180^{o} - \theta_1];$

$sin[\delta_3] = cos[51.215^{o}]cos[76.085^{o}] + sin[51.215^{o}]sin[76.085^{o}]cos[144.930^{o}];$

Thus, $\delta_3 = -27.945^{o}$ (4 points)

Applying sine rule, $VPI = arcsin\left[\frac{sinPVIsinVI}{sinPI}\right] = arcsin\left[\frac{sin144.930^{o}sin76.085^{o}}{sin117.945^{o}}\right] = 39.148^{o}$

Hence, $\alpha_3 = \alpha_1 - VPI = 279.235^{o} - 39.148^{o} = 240.087^{o} \approx 16^h0^m21^s$ (5 points)

Out of 5 points, 1 point is reserved for realising that the intersection point is in past path.

(f) The total velocity along the great circle for Vega is

$\mu_1 = \sqrt{(\mu_{\alpha1}cos\delta_1)^2 + {\mu_{\delta1}}^2} = \sqrt{200.94^2 + 286.23^2} = 349.72$ mas/year (2 points)

Number of years = $\frac{VI}{\mu_1} = \frac{76.085\times3600}{0.34972} \approx 783200$years

It will happen in the year 781200 BCE (2 points)

Similarly, $\mu_2 = 660.30$and number of years =363700 years

It will happen in the year 3617 BCE (2 points)

(g) After 363700 years, Vega would have traversed 35.335º along its path. (3 points)

Thus, its separation from Altair will be (76.085º – 35.335º) = 40.750º (2 points)

(h) In earth centric Cartesian frame, coordinates of Vega (in units of parsec) will be:

$x_1 = r_1cos\delta_1cos(360^{o} - \alpha_1) = 0.96062$pc

$y_1 = r_1cos\delta_1sin(360^{o} - \alpha_1) = 5.90793$pc

$z_1 = r_1sin\delta_1 = 4.80998$pc

Similarly, coordinates of Altair will be, x$_2$= 2.35579, y$_2$= 4.48736 pc and z$_2$= 0.79097 pc.

(6 points)

Here, we are assuming the North as positive x-direction and pole as positive z-direction.

The direction vectors for velocities in spherical coordinates are

$v_1 = (-13.9, 7.33, 10.45)$and $v_2 = (-26.9, 19.57, 14.06)$ (4 points)

The same in Cartesian coordinates will be

Similarly, Cartesian velocity vector of Altair will be, v$_2$= (4.33, -33.85, 9.87). (8 points)

Lastly, we evaluate the value of determinant,

= 279.82 – 65.81 – 286.48 = -72.47

As the value of the determinant is non-zero, the direction vectors of these two stars do not cross each other. Hence no such point can exist. (4 points)